Modelling of Heterogeneously Distributed Soluble boron and Coolant Density within a Fuel Assembly

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Modelling of Heterogeneously Distributed Soluble Boron and Coolant Density Within a Fuel

Assembly

BJÖRN EKLÖF

Master of Science Thesis Division of Reactor Physics, KTH

Supervisors: Janne Wallenius Marcus Eriksson Examiner: Janne Wallenius

TRITA-FYS 2011:64

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iii

Abstract

This thesis will investigate the implications of extending a cross-section model based on flat, homogeneous, boron distribution and density states within a fuel assembly, with correction terms from heterogeneously distributed boron and coolant density. The heterogeneous states refers to differences in active coolant and bypass coolant states (which is not in contact with the fuel). The aim is to verify the implementation of this model in the simulator software POLCA-T, used at Westinghouse to simulate BWR transients, as well as to verify the model for use in transient simulations. This was performed by solving different transient problems and by comparing the results with other cross-section models.

The standard way of modeling the bypass is to average all bypass sections into one collective, averaged bypass. In this thesis, the effect of modeling with multiple bypass channels, and the implications on

heterogeneities, have been investigated.

It was found that in many cases, differences in coolant flow velocities between the active regions and bypass region generate heterogeneous boron states.

These heterogeneities become large enough for the heterogeneous effects on reactivity to be observed. It was also found that in the event of loss of pressure and onset of bypass boiling, heterogeneous density states with noticeable reactivity effects were present. The investigation into multiple bypasses lead to inconclusive results.

The evaluation confirms that the model has been correctly coupled to POLCA-T and performs as intended.

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Acknowledgements

I would like to thank my mentor Marcus Eriksson for his support and invaluable feedback. For his time spent on reading and ensuring a high academic standard on this report, I would like to thank Janne Wallenius at the Division of Reactor Physics at KTH. I would also like to thank Westinghouse Electric Sweden AB in Västerås for allowing me access to their extensive knowledge, industry leading technology and time from their employees, especially Ulf Bredolt who helped me with a lot of bug fixes and implementing new features as needed, and Yonatan Dag for preparational work on the subject.

v

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List of Figures

1.1 Thermal power profile of quarter symmetry ABWR core. . . . . 3

1.2 Topological map of a BWR fuel bundle . . . . 4

1.3 Neutronic cross section for b

¹⁰

. . . . 5

1.4 Neutronic cross section for water . . . . 6

1.5 Neutron cross section data for U

²³⁵

. . . . 7

2.1 Heterogeneous cross term for current- and historic densities. . . . 18

2.2 Heterogeneous boron term for current- and historic densities. . . . 18

2.3 Heterogeneous density term for current- and historic densities. . . . . . 19

2.4 Overview of reduced model. . . . 20

2.5 Bypass mappings of quarter symmetry nuclear reactor core . . . . 22

3.1 Case 1, 300 kg/s boron injected, k

_eff

vs time. . . . 26

3.2 Case 1, 300 kg/s boron injected, boron concentration & concentration difference vs. time, from central active bundle and bypass. . . . 26

3.3 Case 2, circulation flow reduced to 1000 kg/s, k

_eff

vs time. . . . 27

3.4 Case 2, circulation flow reduced to 1000 kg/s, bypass density difference from reference vs. time. . . . 27

3.5 Case 3, 7 kg/s boron injected, circulation flow reduced to 1000 kg/s, k

_eff

vs time. . . . 28

3.6 Case 3, 7 kg/s boron injected, circulation flow reduced to 1000 kg/s, boron concentration & concentration difference vs. time, from central active bundle and bypass. . . . 28

3.7 Case 3, 7 kg/s boron injected, circulation flow reduced to 1000 kg/s, bypass density difference from reference vs. time. . . . 29

3.8 Case 4, 300 kg/s boron slowly injected, k

_eff

vs time. Long runtime. . . . 29

ix

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equal size bypass. . . . 33

3.17 Case 6, circulation flow reduced to 1000 kg/s, ρ

_bp

vs. time. 5-channel bypass. . . . . 34

3.18 ATWS power comparison . . . . 34

3.19 ATWS k

_eff

comparison . . . . 35

3.20 ATWS cladding temperature comparison . . . . 35

3.21 ATWS containment temperature rise comparison . . . . 36

3.22 ATWS boron distribution . . . . 36

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List of Tables

A.1 Calculation of fuel costs for 1 kgU enriched fuel in units of thousand US$. 47

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Chapter 1

Introduction

The importance of simulators and their applications in science and engineering have grown substantially over the last decades. Driven mainly by hardware improve- ments, faster computers have made more complex simulations possible, relying less and less on simplifications and assumptions on symmetry for acceptable run times.

In the context of nuclear engineering simulators have been used for a long time, mainly to predict fuel burn and to optimize fuel economy, but also for safety anal- ysis. Fuel optimization is a matter of improving cost efficiency and has historically driven development towards more complex software. With fuel costs currently at

∼ e100 per kg UF

6

[1], resulting in an end price of around e1800 per kg enriched Uranium (see appendix A.1); the incentive to maximize burnup is more important than ever. By taking storage and handling of the waste products into considera- tion, the incentive to minimize volume becomes even stronger. In later years safety concerns have risen in importance, wherein transient analysis plays a central role as malfunctions give rise to chains of events which are intrinsically transient, and so the analysis must go beyond steady state.

This thesis will investigate the effects and properties of implementing a model for heterogeneously distributed boron and density in the context of transient simula- tions in Westinghouse’s transient simulation code POLCA-T.

1.1 Aims and Objectives

The aim is to verify the coupling of the heterogeneous cross-section model to POLCA-T. Analysis is to be performed to verify that the model gives results

1

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eff

neutrons)/#(absorbed + leaked neutrons). A value greater than one signifies rising neutron density and increasing number of fissions per second, i.e. a supercritical core. A value less than one signifies decreasing neutron density and fewer fissions per second, i.e. a sub-critical core. A value of exactly one means an equilibrium core, with the same number of neutrons produced as used.

Reactivity coefficients:

Reactivity is a macroscopic measurement, which can be analyzed on a local or global scale. The reactivity depends on the macroscopic cross section, which is made up by reactivity coefficients contributing positive or negative reactivity depending on whether the phenomena described increases or decreases the macroscopic cross sec- tion. The reactivity coefficients thus are related to the cross section models and are by no means physical constants.

1.3 Reactor Model

The reactor simulated is the Westinghouse Advanced Boiling Water Reactor (ABWR), an internal recirculation pumps BWR with a rating of 3,93 GW

_th

deployed with SVEA Optima-2 fuel loaded as an equilibrium core. The reactor core has 872 fuel bundles and 205 control rods. The nominal specific power density is 25,3 kW/kgU.

The simulations carried out in this report are at the end of the fuel cycle with an average exposure of 37,0 MWd/kgU.

1.3.1 Bypass

The fuel assembly simulated is the SVEA-96 Optima2 fuel. It consists of four cells

containing 24 fuel pins each, arranged on a square. The coolant in direct contact

with the fuel pins, inside the cells, is defined as the active coolant. The flow paths

between the cells, as well as between the assemblies, are defined as bypass regions.

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1.4. BORON DEPLOYMENT IN NUCLEAR REACTORS 3

Figure 1.1: Thermal power profile of quarter symmetry ABWR core.

Within each assembly there is one bypass water cross (WX) in the middle and four water wings, separating the cells. These five bypass regions are collectively called the intra-assembly bypass. Between the assemblies lays the inter-assembly bypass.

The bypasses are not in direct contact with the fuel and act as a neutron moderator and thermal buffer.

When simulated, all bypass regions for each assembly are added to a lumped assem- bly bypass, marked red in figure below. The standard method is then to average all lumped assembly bypasses into one effective bypass channel. An investigation into mapping the assembly bypasses into several channels laid out in radial regions has been performed.

1.4 Boron Deployment in Nuclear Reactors

The boron cross section mentioned in the section above is for the case of emergency shut down of the nuclear reactions. If the control rods can’t deploy, boron tanks are present to pump borated water directly to the upper plenum, right above the core. The system can either be manual or automatic, triggered by sensors and event chains.

The reason for boron being used for this purpose is due to its large cross section,

and thus its ability to absorb thermal neutrons efficiently and making the reactor

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Figure 1.2: Svea-96 Optima2 fuel assembly. Bypass regions marked red.

sub-critical. The isotope B

¹⁰

has a high cross section, almost 10

⁴

barns for thermal neutron (see figure 1.3), compared to the naturally more abundant B

¹¹

, with cross sections of about 5 barns for thermal neutrons [9].

The nuclear process of neutron absorption for B

¹⁰

[4]:

B

¹⁰

+ n −→ B

¹¹_∗

Li

⁷

+ He

⁴

+ γ + 2,4 MeV (1.1) In typical cases of boron injection the concentrations reaches a couple of hundred ppm’s (relative weight), which equates to number densities

¹

of ∼ 10

²⁷

m

⁻³

. This can be compared to the typical neutron number density

²

of ∼ 10

¹⁹

m

⁻³

, which ensures that there is more than enough boron present to stop reactions and no risk for the boron to burn out.

The fact that water has a cross section of about 50 barn compared to 10

⁴

barn for B

¹⁰

for thermal neutrons, seen in figure 1.3 & figure 1.4 (together with number densities of 6,24 · 10

³⁰

m

⁻³

for water and 10

²⁷

m

⁻³

for B

¹⁰

), ensures that the boron

1Concentration is defined as mass relatives, but since the mass difference between boron and water is ∼ 2 and only order of magnitudes are compared, with saturated liquid water density under operation: ∼ 740 kg/m³⇒ 6,24 · 10³⁰m⁻³.

2power = fissions/s · energy/fission ⇒ fissions/s = power/(energy/fission). Fissions/s ' 4 GWth/200 MeV = 2,5·10²⁸/2·10⁸s⁻¹= 1,14·10²⁰s⁻¹⇒ total number neutrons (at equilibrium)

= 2,5 · 1,14 · 10²⁰≈ 3 · 10²⁰. The neutron number density then becomes (with nuclear core volume of ∼ 64 m³): ∼ 5 · 10¹⁹ n/m³.

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1.5. VOID INTERACTION 5

has high chances of interaction with the neutrons; The quotient of the water and the boron products of number density and cross section gives a simplified relative probability for interaction with the neutrons, which in this case is ∼ 0,1 for boron, or about a tenth the probability to interact with water.

The numbers stated above are for a BWR (Boiling Water Reactor). In BWR’s natural boron is used, with an isotope distribution of 80,18 % B

¹¹

and 19,82 % B

¹⁰

[2], and thus only about 20 % of the mass contributes significantly to the macro- scopic cross section. The situation is slightly different for PWR’s (Pressure Water Reactor), since they operate under much higher pressures they need higher concen- trations of active boron to reach adequate probabilities of interaction with thermal neutrons. To reach the higher concentrations, boron enriched with B

¹⁰

is deployed instead of natural boron.

Figure 1.3: Neutronic cross-section for B

¹⁰

[8].

1.5 Void Interaction

Void fraction, or percentage of coolant in gas phase, generally has a strong impact on

reactivity. Since almost all commercial nuclear reactors are thermal reactors where

the main absorption is of moderated neutrons with kinetic energy comparable to

the thermo-dynamically available energy (thermal neutron: E

_kin

' k

_B

T ' 0,025 eV

). Since neutrons produced from nuclear reactions typically have kinetic energies

of ∼ 2 MeV [7], they need to be moderated (slowed down) to thermal energies to

increase the absorption probability, according to figure 1.5. If the neutrons can’t

be slowed down, the reaction probability will be low and many neutrons will leak

out of the nuclear core. By escaping the core, the neutrons available to initiate

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Figure 1.4: Neutronic cross section for liquid water and the sum of its components [6].

new fissions decreases and if the conditions persist the nuclear rector will become sub-critical.

When void levels increase, the general density of the coolant decreases. Since the

coolant acts as the moderator in light-water reactors, the general loss of mass density

means a general loss of number density. With less water molecules present, the

probability of a neutron to strike a molecule and become moderated decreases,

which in turn leads to a longer mean free paths and a higher probability of escaping

the nuclear core. This is one of the main safety features in BWR’s, since if there

is a problem with the coolant flow, this will lead to loss of pressure which gives a

higher void fraction, which in turn slows or shuts down the reactor. In PWR’s, it is

true to some extent that void interaction acts as a safety feature, in the sense that

in case of a coolant leak this will lead to voiding which will lower the reaction rate.

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1.5. VOID INTERACTION 7

Figure 1.5: Neutron cross section data for U

²³⁵

[5]

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Chapter 2

Methodology

2.1 The Simulator, POLCA-T

POLCA-T is a coupled computer code for transient thermal-hydraulics and neutron kinetics analysis of BWR and can be used as a general tool for advanced simula- tion of single and two-phase flow systems including none condensable gases. The code consists of POLCA7, a steady state solver for a given operating point, which in turn is introduced to a discretizised time line by POLCA-T. Execution of the steady state solver is performed repeatedly for the operating point the system re- sides in, utilizing the output of the previous solution as the input for the next time step. The code has a full 3D model of the reactor core. The reactor pressure ves- sel, external pump loops, steam system, feed water system, ECC (Emergency Core Coolant) systems and steam relief system can be modeled to the desired details.

Control and safety systems are modeled using a code package called SAFIR, which is linked to the code. The SAFIR package is in production use together with the 1D BWR transient code BISON.

The application areas of the POLCA-T cover operational transient, stability, RIA (Reactivity-Initiated Accident), ATWS (Anticipated Transients Without Scram), ATWC (Anticipated Transients Without Control rods) and LOCA (Loss Of Coolant Accident).

The neutronics models in the code are the same as those in the static nuclear core analyzer POLCA7 with addition of proper kinetic terms for transient use. The thermal-hydraulic model has advanced physics with thermal non-equilibrium and full geometric flexibility.

The POLCA-T code is well adapted to analyze the type of scenarios with a number

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3D two group kinetics model allowing for up to six delay neutron groups calculates the fission power generation in the core. Reactivity feedback is included for void fraction, moderator (coolant) temperature, fuel temperature, and reactor control rods. The power fraction deposited in the fuel and coolant can be specified versus time or as a function of coolant density.

The decay power generation is calculated by the sum of eleven fission product decay groups and the actinide decay of Uranium-239 and Neptunium-239 or by a user provided table as decay power versus time.

2.1.2 Cross sections in POLCA7

POLCA7 relies on homogenized (spatially smeared) two-group macroscopic and mi- croscopic cross sections, diffusion coefficients and discontinuity factors to solve the neutron diffusion equation. The generation of the nodal data and their tabulation as functions of local physical states is done outside of POLCA7 by means of two- dimensional (2D) lattice physics calculations. Nodal data is generated for individual fuel segment types which is uniquely defined by its lattice design (geometry, enrich- ment, burnable absorber, loading, e.t.c.). The nodal data is generated as a discrete set of cross sections as functions of state parameters. The cross section for a given state point is then interpolated linearly from the closest tabulated points.

2.2 Multi Group Theory

Cross sections dictates absorption probabilities, and thus is a governing factor of reactivity. In nuclear reactors the reactivity is important for determining criticality, power generation and general dynamics of the core. The size of the cross sections depends on several factors, incident particle energy being among the most impor- tant.

Figure 1.3 & 1.4 show examples of how the cross sections are dependent on incident

particle (in this case neutron) energy.

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2.2. MULTI GROUP THEORY 11

In designing POLCA-T, the balance between precision and the work needed for generating cross sections was chosen to be two groups for active neutrons, meaning that particles (neutrons) belong to either a fast or a thermal energy group. Delayed neutrons is represented by six groups and decay power from eleven groups. For each group separate cross sections for all different materials and state point combinations have to be calculated.

Neutronics is handled by POLCA7 through representation (for each group) by num- ber densities as functions of position and time. In each time step, the neutronic solver calculates the change in number density for each group in each discrete point, from diffusion, absorption and flows between the groups (e.g. one way flow towards lower energy). The rate of change of neutrons, multiplied with the number density gives the neutron flux for each spatial point.

Absorption rate depends on the cross section of each material present, for each

energy group as well as the composition of materials at each point. Diffusion de-

pends on the neutron density and gradient in a given point. Group transitional flow

depends on transition probabilities of the materials present and neutron number

density of the high energy group at a given node. The mathematical formalism is

explained in detail in [3].

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2.3.1 Standard POLCA7 cross section Model

The standard POLCA7 cross section model is based on a combination of a base cross section and deviation terms. The base cross sections are valid at base conditions, a predefined set of states, not including exposure (E), historic coolant density (ρ

_h

) or current active coolant density (ρ). Cross sections for different state points are then interpolated from the nearest set of base conditions. The cross sections (both macroscopic and microscopic) takes the form:

Σ = Σ

^b

(E,ρ,ρ

_h

) + ∆Σ

^CR

(E,ρ,ρ

_h

) + ∆Σ

^SG

(E,ρ) + d

_dop

(E,ρ)

p

T

_f

−

^q

T

_f^b

+ +b

_B

(E,ρ,ρ

_h

) [C

_B

− C

_B

] + X

_Xe

(E,ρ,ρ

_h

) [N

_Xe

− N

_Xe

(E,ρ

_h

)] +

+Σ

_i

σ

i

(E,ρ,ρ

_h

,CR, T

_f

, N

_Xe

, C

_B

)[N

_i

− N

_i^b

(E,ρ

_h

)]+

+ H.O.T. + Cross terms + Intra − nodal correction terms (2.1) Where the terms account for control rods (CR), spacer grids (SG), Doppler effect (dop), soluble boron concentration (C

_B

) and Xenon (Xe) as well as heavy nuclide isotopic corrections. T

_f

denotes the node average fuel Doppler temperature and C

_B

the average soluble boron concentration in units of (ppm).

Cell data in general are computed as functions of exposure (E), historic coolant

density (ρ

_h

) and current active coolant density (ρ). During generation bypass wa-

ter regions are set to correspond to the active coolant liquid density, i.e. a flat,

homogeneous state. The boron coefficients are computed based on homogeneous

(or flat) distribution throughout the assembly. The governing concentration when

using the cell data is the boron concentration of the active coolant.

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2.3. MODELS DESCRIPTIONS 13

Regarding functional dependence of the fuel Doppler effect

The square root fuel temperature dependence seen in equation 2.1 is not a fun- damental physical relation of the Doppler effect, but rather a coincidence. For the neutron spectrum in thermal LWR’s, the temperature dependence of the Doppler ef- fect coincides with the square root function. For most parts of the kinetic spectrum this is not the case however, instead the temperature dependence have to be curve- fitted to experimental data and in general one can only say that the temperature dependence is proportional to a fractional exponent of temperature.

2.3.2 Effective Coolant Density Model

This model uses the standard cross section model with the difference that an effec- tive coolant density is used to look up cross sections in the cell data tables. This means that the coolant density ρ is replaced by an effective density, ρ

_eff

. The effec- tive coolant density model will conserve the total coolant mass in the active region plus the bypass region. The effective density applied in the cell data interpolations takes into account the deviation in the calculated bypass density from the reference bypass density used during cross section data generation. By default the ρ

^ref_bp

is set to ρ

^liq,sat_bp

, a reference density for liquid phase coolant. The model is derived from:

ρ

_act

A

_act

+ ρ

_bp

A

_bp

= ρ

_eff

A

_act

+ ρ

^ref_bp

A

_bp

⇔ ρ

_eff

= ρ

_act

+ A

_bp

A

_act

(ρ

_bp

− ρ

^ref_bp

) (2.2) Where A

_act

is the active coolant flow area, A

_bp

is the bypass flow area, ρ

_act

is the current active coolant density, ρ

_bp

is the bypass density.

2.3.3 Effective bypass-boron Model

This model uses the standard cross section model with the difference that an effective boron concentration is used to look up cross sections in the cell data tables. The boron concentration C

_B

is replaced by an effective concentration C

_B^eff

. The effective boron will conserve total boron concentration in the active region plus bypass region.

The effective density applied in the cell data interpolations takes into account the deviation in bypass boron concentration used during cross section data generation.

By default the bypass boron concentration is set to be identical to the active coolant

boron concentration.

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+ steam phase, ρ

_bp

is the liquid + steam phase density in the bypass channel.

2.3.4 Heterogeneous Model

The heterogeneous model uses specifically generated cell data to account for het- erogeneous states. By doing so, assumptions are made that differences in bypass density or boron concentration states affects the cross sections the same way as the same difference in active coolant. Both effective models make this assumption by only scaling the deviations with the bypass fractional area.

As with any linear interpolation, the further from the generation point one gets, the more the error tends to grow. Therefore the heterogeneous model should provide more accurate results especially at states further away from generated points. The next section describes the heterogeneous model in depth.

2.4 The Heterogeneous Model in Depth

2.4.1 Aim & Purpose of the Model

The heterogeneous model aims to improve the accuracy of which the cross section modeling in POLCA-T. In the event of voiding in the bypass or injection of borated water, the conditions assumed during cell data generation are no longer valid. Since the bypass makes up about 1/3 of the coolant flow area of the fuel assembly, any slight changes to state here will have a noticeable impact on the total macroscopic cross section. In the context of transient simulations, the heterogeneous model is an attempt to treat reactivity feedback originating from coolant voiding in the bypass and/or presence of soluble boron in greater detail.

The contribution to the macroscopic cross section is given by the expression:

∆Σ

^het

(E, ρ

_his

, ρ) = b

^byp_B

(E, ρ

_his

, ρ) ·

C

_B^byp

− C

_B^act

+

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2.4. THE HETEROGENEOUS MODEL IN DEPTH 15

+b

^byp_ρ

(E, ρ

_his

, ρ) ·

ρ

^byp

− ρ

^act,ref_liq

+

+ b

^byp_B,ρ

(E, ρ

_his

, ρ) ·

C

_B^byp

− C

_B^act

ρ

^byp

− ρ

^act,ref_liq

(2.4) Where b

^Y_X

is the tabulated cross section data, E is burnup, ρ

_his

is historic density in kg/m

³

, ρ

^act,ref_liq

is the active channel reference density for saturated water, ρ

^byp

is bypass instantaneous density in kg/m

³

and C

_B

is boron concentration in ppm for active and bypass channels respectively.

2.4.2 Void Contribution

In the event of voiding in the bypass, there will be a change in the macroscopic cross section due to loss of moderating water molecules (due to lowered density).

It is currently assumed that by using cross section data generated without bypass void, conservative calculations are performed since this assumption leads to higher reactivity; if there are margins to acceptance criteria with this assumption, there will be even larger margins if void is present in the bypass. The inclusion of bypass void corrections to reactivity will give more accurate results and will increase un- derstanding of actual behavior under such conditions. From equation 2.4, it can be seen that the void correction will increase reactivity if the bypass has higher density than the reference value used in generating the standard cross section data, and decreased reactivity if the density is lower than the reference value. The correction therefore compensates the cross- section if density differs from reference, and the assumption made in the standard method no longer holds.

2.4.3 Boron Contribution

The standard method for simulating injection of borated water is to assume equal concentrations in the active fuel channels and its surrounding bypasses. Since the boron transport in POLCA-T is based on local mass-balance, the boron flux depends mainly on the flow rate of the coolant in each region. Unless the flow velocity of the coolant in liquid phase, as well as the boundary conditions, are equal for both the active channel and its bypass, concentration gradients will emerge over the active channel and its bypasses, i.e. a condition of heterogeneous boron distribution. Since the diffusion of boron is temporal, it is a dynamic, transient process. When the boron has diffused and reached a condition of equilibrium, the heterogeneous effect vanishes. Thus as long as there is no rapid change in boron concentration fed to the reactor core, a homogeneous boron concentration distribution will be present and the standard cross section model is accurate.

The effect of injecting borated water with the heterogeneous model is such that if

the boron reaches the bypass first, the boron will have higher efficiency compared

to the standard cross section model. The physical explanation being that under

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the size of the coefficient is three orders of magnitude smaller than the first order corrections, at its largest value (which is seen in figures 2.1, 2.2 & 2.3), and thus will for most cases have a very small contribution, overwhelmed by the first order corrections. Any such contribution will be close to impossible to identify in any global state parameters such as k

_eff

or the thermal power, therefore no investigation into this specific term will be performed in this report. An illustrative example is given below showing the needed magnitudes of the perturbations to make this second order term of significant size. Only thermal neutrons are compared in the example, since the values for fast neutrons are two orders of magnitude smaller for all coefficients, thus their contribution to a global parameter such as k

_eff

will be significantly smaller. Furthermore the magnitudes for the fast neutron coeffi- cients have the same distribution as the ones for thermal neutrons (i.e. 2 orders of magnitude at its largest), therefore any claims to the relative importance made for thermal neutrons should be possible also for fast neutrons.

A simple example highlighting this follows, with cell data as:

BDHSA2 = 4,63 · 10

⁻¹⁰

, BHSGA2 = 6,83 · 10

⁻⁶

, DHSGA2 = −6,29 · 10

⁻⁷

Where BDHSA2 is the cross-correlation coefficient, BHSGA2 the boron coefficient and DHSGA2 the density coefficient for the heterogeneous part of the macroscopic thermal absorption cross section.

These data are from an active volume cell of channel 202 (i.e. an active chan- nel close to the center of the core) in the ABWR, with conditions ρ

_h

=ρ=460 kg/m

³

and mean burn up of 40 MWd/kgU, which are typical values for active channels under normal operation (38 % void).

4,63 · 10

⁻¹⁰

x

²

≥ 7,51 · 10

⁻⁶

x

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2.4. THE HETEROGENEOUS MODEL IN DEPTH 17

Here, the variable has been defined in favor of the cross-term having more influ- ence. With this inequality, one can show for what value of the variable x the cross term exceeds a preset condition (where x is the state variables of interest: boron concentration difference and density deviation from reference value).

For the cross term to grow to at more than 10 % of the total correction term, we look at the inequality:

4,63 · 10

⁻¹⁰

7,51 · 10

⁻⁶

x ≥ 0,1 ⇔ x ≥ 1,62 · 10

³

Where the most favorable conditions for the cross term has been assumed (i.e.

concentrations and density differences of the same magnitude). The result shows that concentrations and density differences from reference of the magnitude of about 1000 ppm boron concentration difference, as well as about 1000 kg/m

³

in density difference, need to be present at the same time to make the cross term contribute more than 10 % of the total correction. Such large concentration differences as well as density differences are not present, at the same time, under realistic conditions.

For the condition that the cross term should contribute at least 1 %, the same

calculations give that magnitudes of 100 for both variables are needed at the same

time. When boron is injected the fission reactions stop abruptly, meaning that

the void will decrease and thus situations where both conditions are present at the

same time are rare. This in addition to the conservative assumptions means that for

realistic values, the cross term only contributes less than 1 % of the total correction,

which in turn confirms the previous statement, that the term is too small to analyze

exclusively. As was discussed before the example, in a state where the cross-term

is most important, i.e. where BDHSA2 is largest compared to when BHSGA2 is

smallest (from figure 2.1 and 2.2), the above calculations have to be scaled with

about 0.5; i.e. magnitudes of 800 or 80 for 10 % and 1 % contribution respectively.

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Figure 2.1: Values for BDHSA2 (Boron & density cross coefficient) for current- and historic density state parameters, with burnup of 40 MWd/kgU.

Figure 2.2: Values for BHSGA2 (Boron coefficient) for current- and historic density

state parameters, with burnup of 40 MWd/kgU.

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2.5. SYSTEM MODEL 19

Figure 2.3: Values for DHSGA2 (Density coefficient) for current- and historic den- sity state parameters, with burnup of 40 MWd/kgU.

2.5 System Model

The model used for verification is the Westinghouse ABWR, described in section 1.3. For analytic purposes, modification to the predefined system model will be covered as well.

2.5.1 Reduced System Model

All the transients run for purpose of the model verification were performed with

a reduced version of the full ABWR, compromising only of the core and the two

closest upper and lower plenum sections. The boundary conditions chosen used

were a constant pressure boundary at the upper plenum, and a constant circulation

flow at the lower plenum, set to the full power steady state values of the full ABWR

model. Figure 2.4 shows the schematics (with three active channels visible) of the

reduced model. The reason for reducing the model is to make sure that effects noted

are not caused by feedback and/or disturbances from the external systems, as well

as for shorting the run times and lower the tendency for diverging solutions.

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approach, one average bypass channel is treated. This bypass channel exchanges

Figure 2.4: Overview of reduced model. Three active channels showing (out of 218

for quarter symmetry). The bypass and the external bypass are also showing, as

well as the boron container near the top left.

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2.6. TEST CASES 21

heat with all (active) fuel channels throughout the assembly walls and internal wall surfaces. Heat is also directly deposited in the bypass by neutron and gamma radiation. As part of the increased focus on the bypass with the heterogeneous model, an investigation into modeling multiple bypasses were performed. With the heterogeneous model, the cross sections are directly influence by heterogeneous states within the bypass, why it is important to investigate if using several bypasses changes the results; since heterogeneous states can be local and thus only affect local neutronics, this can have implications for the results.

The following bypass mapping configurations were studied:

1. Two internal bypasses mapped similar to the throttle map.

2. Two internal bypasses with almost equal flow areas.

3. Five internal bypasses with almost equal flow areas and with outermost region similar to throttle map.

Due to convergence issues, the bypass map visualized in figure 2.5b was chosen not to strictly follow the throttle zone. By doing this, a slight averaging occurs due to mixing of sources of two different flow velocities into one channel. This channel will see a mass flow which lies in-between the mass flows of the two sources by themselves, thus reducing the strain on the boundary region between the two bypass channels modeled. The chosen mappings were tested to investigate what influence the mapping has to the simulation results, and to verify that there are no scaling errors or other partitioning effects on the results. In all cases of multiple bypass channels, there are no interconnections between the bypass channels, meaning that no transversal flow, coolant water or heat, can occur except at the upper and lower plenum sections.

2.6 Test Cases

The four models presented in section 2.3 is the only difference between the results within each test case. All other parameters such as disturbance, initial state and boundary conditions are identical.

2.6.1 Disturbances

The cases run were the following disturbances, for a single bypass configuration:

1. Borated water injected to lower plenum with a flow of 300 kg/s. Start from 0

kg/s, then at 1 s ramping up reaching 300 kg/s at 2 s, continuing at 300 kg/s

for the rest of the run. Coolant flow kept at nominal value of 14500 kg/s.

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(a) Throttle zones of reactor core. (b) Bypass map with two channels, mapped after throttle zones.

(c) Bypass map with two channels, approximately equal flow area for both channels.

(d) Bypass map with five channels, approximately equal flow area for all channels.

Figure 2.5: Bypass mappings of quarter symmetry nuclear reactor core, seen from

above.

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2.6. TEST CASES 23

2. Coolant flow reduced to 1000 kg/s. Start from 14500 kg/s, then at 1 s ramping down to 1000 kg/s at 6 s, continuing at 1000 kg/s for the rest of the run.

3. Borated water injected to lower plenum with a flow of 7 kg/s. Start from 0 kg/s, then at 1 s ramping up reaching 300 kg/s at 2 s, continuing at 7 kg/s for the rest of the run. Coolant flow reduced to 1000 kg/s. Start from 14500 kg/s, then at 1 s ramping down to 1000 kg/s at 6 s, continuing at 1000 kg/s for the rest of the run.

4. Identical to case 1, but longer run. Borated water injected to lower plenum with a flow of 300 kg/s. Start from 0 kg/s, then at 1 s ramping up reaching 300 kg/s at 300 s, which is end of run. Coolant flow kept at nominal value of 14500 kg/s.

For multiple bypasses (2 bypass channels mapped after throttle zones, 2 bypass channels with equal flow area and 5 bypass channels with equal flow area) all runs have the heterogeneity model enabled. The following cases were run:

5. Borated water injected to lower plenum with a flow of 300 kg/s. Start from 0 kg/s, then at 1 s ramping up reaching 300 kg/s at 2 s, continuing at 300 kg/s for the rest of the run. Coolant flow kept at nominal value of 14500 kg/s.

6. Coolant flow reduced to 1000 kg/s. Start from 14500 kg/s, then at 1 s ramping down to 1000 kg/s at 6 s, continuing at 1000 kg/s for the rest of the run.

2.6.2 A Realistic ATWS Simulation

To test the performance outside synthetic, simplified test cases, a test run on the full ABWR ATWS was performed. This is the model from which the reduced, simplified model was taken for test runs previously mentioned. Two transients were run, one with the standard model and one with the heterogeneous model enabled.

The simulations are run from the same steady state as in the reduced model. The purpose of run is to observe changes in stability and convergence in of the model, and to see if the contribution changes values relevant to acceptance criteria.

The ATWS transient is a case where the reactions are to be halted, but where

the control rods do not deploy. In such an event a predefined routine is run, where

circulation flow is reduced and boron is automatically injected into the nuclear core.

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Chapter 3

Results

HetXS refers to the heterogeneous model in the figure legends below.

3.1 Transient Simulations

Below follows results of the transient simulations, presented case-wise.

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Figure 3.1: Case 1, 300 kg/s boron injected, k

_eff

vs time.

Figure 3.2: Case 1, 300 kg/s boron injected, boron concentration & concentration

difference vs. time, from central active bundle and bypass.

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3.1. TRANSIENT SIMULATIONS 27

Case 2

Figure 3.3: Case 2, circulation flow reduced to 1000 kg/s, k

_eff

vs time.

Figure 3.4: Case 2, circulation flow reduced to 1000 kg/s, bypass density difference

from reference vs. time.

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Figure 3.5: Case 3, 7 kg/s boron injected, circulation flow reduced to 1000 kg/s, k

_eff

vs time.

Figure 3.6: Case 3, 7 kg/s boron injected, circulation flow reduced to 1000 kg/s,

boron concentration & concentration difference vs. time, from central active bundle

and bypass.

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Figure 3.7: Case 3, 7 kg/s boron injected, circulation flow reduced to 1000 kg/s, bypass density difference from reference vs. time.

Case 4

Figure 3.8: Case 4, 300 kg/s boron slowly injected, k

_eff

vs time. Long runtime.

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Figure 3.9: Case 4, 300 kg/s boron slowly injected, average total boron concentra- tion vs. time. Long runtime.

3.1.2 Multiple Bypass Case 5

Figure 3.10: Case 1 & 5, 300 kg/s boron injected, k

_eff

vs. time. Multiple bypass.

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Figure 3.11: Case 5, 300 kg/s boron injected, (C

_B

)

_bp

vs. time. 2-channel bypass.

Figure 3.12: Case 5, 300 kg/s boron injected, (C

_B

)

_bp

vs. time. 2-channel equal size

bypass.

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Figure 3.13: Case 5, 300 kg/s boron injected, (C

_B

)

_bp

vs. time. 5-channel bypass.

Case 6

Figure 3.14: Case 2 & 6,circulation flow reduced to 1000 kg/s bypass density dif-

ference from reference vs. time. Multiple bypass.

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Figure 3.15: Case 6, circulation flow reduced to 1000 kg/s, ρ

_bp

vs. time. 2-channel bypass.

Figure 3.16: Case 6, circulation flow reduced to 1000 kg/s, ρ

_bp

vs. time. 2-channel

equal size bypass.

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Figure 3.17: Case 6, circulation flow reduced to 1000 kg/s, ρ

_bp

vs. time. 5-channel bypass.

3.2 Full Scale ABWR ATWS

Figure 3.18: ATWS power comparison. Red is with the heterogeneous model, green

without.

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3.2. FULL SCALE ABWR ATWS 35

Figure 3.19: ATWS k

_eff

comparison. Red is with the heterogeneous model, green without.

Figure 3.20: ATWS cladding temperature comparison. Red is with the heteroge-

neous model, green without.

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Figure 3.21: ATWS containment temperature rise comparison. Red is with the heterogeneous model, green without.

Figure 3.22: ATWS boron distribution vs. time. With heterogeneous model.

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Chapter 4

Discussion of Results

The results obtained are in the form of plots, presented in chapter 3. The system complexity makes any theoretical prediction of significance impossible.

4.1 General

To verify computations one usually refers to an equivalent computation performed, either by simplified theoretical arguments, or by simulations or with simulation code known to be accurate. The problem with verifying transients is that a reference so- lution seldom exists; which is the case for the transients studied in this report, with no other transient simulation software available for comparison.

The fact that the heterogeneous model should have a noticeable effect on k

_eff

can be visualized by comparing the magnitude of the macroscopic cross section contri- butions of boron absorption of thermal neutrons for base conditions and the hetero- geneous contribution. By choosing a specific volume cell, record the state (density, density history and burnup) and then compare the magnitude of the heterogeneous boron contribution with the standard model boron contribution, one can see under what conditions the heterogeneous contribution becomes significant. At any such time there should be a noticeable difference in k

_eff

.

The boron coefficients for generating macroscopic thermal absorption cross sections for a typical

¹

, core node are:

BHSGA2 = 0,683 · 10

⁻⁵

, BSIGA2 = 1,01 · 10

⁻⁵

1Typical for the simulations performed, with SVEA-96 Optima2 fuel, with burnup of 40 MWd/kgU, historic- and current density of 460 kg/m³.

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than in the active channel the boron cross section will get a positive contribution, increasing the effect of boron as compared to the standard homogeneous treatment.

On the other hand if there is a higher concentration of boron in the active channel, the opposite will happen and boron efficiency will decrease.

One basic assumption that is made in the modeling of boron transport in POLCA-T is that physical diffusion and turbulent mixing of boron is negligible. Boron trans- port is driven by convective flow alone. In addition, the model assumes that boron is present only in the liquid phase as the vapor pressure of the boron compounds is too low to expect any boron to be present in the steam. Thus, the flow of boron is treated as a slug flow, being carried with the liquid. The liquid flow velocity may very well be different inside the active channel than the water in the bypass re- gion, and this will produce concentration differences between the active and bypass channels. The flow velocity is determined by a numbers of factors such as channel pressure drop, two-phase flow, coolant density, channel orificing, which all have an effect on liquid flow velocity. In the current test model, the flow velocity in the bypass, at steady-state, is lower than in the active channel. Smaller openings in the bypass channel produces higher pressure drops and lower flow velocities, and thus the boron enters the bypass channel at a slower rate.

4.2 Single bypass

In figures 3.1 & 3.2, depicting case 1, the boron enters through the active channels

first, since the boron is injected from below in the reduced model used for verifi-

cation purposes, and since the active channels draws more coolant from the lower

plenum than the bypass, due to higher flow velocities in the active channels. This

means that on average, less boron will be present in each fuel assembly (active and

bypass together) than what is assumed in the standard method (where it is assumed

that the same concentration is present in both active and bypass). With heteroge-

neous model, this concentration difference will change the total macroscopic cross

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4.2. SINGLE BYPASS 39

section; in this case lowered, since there is less boron in the assembly than the assumption made in the standard method. This leads to a situation where boron efficiency is reduced with the result that the reactivity starts to increase after an initial decrease. The largest positive reactivity insertion rate occurs when the con- centration difference is at its largest value, which is in accordance with what is expected from observation in equation 2.4 When there is more boron in the active channel than in the bypass, a positive correction term is added to k

_eff

. Once the ini- tial spatial distribution has settled and a homogeneous state has been reached, k

_eff

of the heterogeneous model model falls and levels out close to the standard model’s steady state k

_eff

. These results indicate that the effects of the boron contribution of the heterogeneous model model are temporal and that the boron contribution from heterogeneous model vanishes in steady state conditions.

The effects of a reduction in core coolant re-circulation flow are illustrated in figures 3.3 & 3.4. The effective coolant density model and the heterogeneous model model are almost identical and the k

_eff

of the standard model lies slightly lower than the other models down at the minimum, but when k

_eff

increases, the standard model lies above the effective model and heterogeneous model. This behavior can be explained by the fact that both heterogeneous model and the effective void model are subject to a negative reactivity contribution from the bypass void. From figure 16 one can conclude that early in the transient the heterogeneous model and effective void models will experience a positive reactivity feedback. After about 8 s, this quantity changes sign and starts to give a negative contribution instead. These effects are also seen in figure 3.5 where the Standard model (neutronically unaware of bypass density) has lower reactivity than heterogeneous model and effective void, but once the bypass density falls below the reference density the standard model has a higher reactivity.

For case 3 with disturbance in both boron and lowered circulation flow, seen in figures 3.5 to 3.7, the combination of the results previously discussed can be noticed.

The explanation for the large heterogeneity seen in boron is that the active channel 202 is used for comparison, and not the general boron levels of the active channels vs. the bypass. The active channel 202 has a central placement and thus is strongly affected by the lowered circulation flow. The fact that k

_eff

for the effective model and the heterogeneous model converge at the end of the figure is caused by the heterogeneities in boron and void act in opposite directions and are of approximately equal size, such that the heterogeneous terms cancel.

To demonstrate that a heterogeneous boron distribution is a transient effect, one

long transient was run with a slowly increasing boron concentration, seen in figures

3.8 & 3.9. As can be seen, heterogeneous model then behaves much like the standard

model, confirming that in slowly evolving transients the boron distribution is much

more uniform.

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to better simulation result.

It would seem that by a finer partitioning, the crest seen in the run with a single bypass decreases in size as well as the transient effect being more stretched in time.

Looking at figures 3.11 through 3.13, depicting the averaged bypass boron concen- trations for the different multiple bypass mappings together with the average bypass boron concentration of the single bypass mapping, it can easily be seen that the two channel mapping, mapped after the throttle zones, follows the concentration of the single bypass mapping closely. This together with the fact that higher boron concentrations reach the more active channels with a split bypass channel (about the same average concentration as a single channel, but distributed with higher concentrations at the active parts) leads to an even more heterogeneous effect on k

_eff

, something that can be seen in figure 3.10. The boron concentrations for the other mappings seem to differ slightly from the single bypass case, with the 5 bypass channel mapping increasing slightly faster than the single bypass early on in the transient, and then slower in the second half, as seen in figure 3.13. The average boron concentration of the 2 bypass channels with equal flow areas is slower as well, which can be seen in figure 3.12. Since it takes longer for boron to reach every- where in the core with these mappings, one understands why in figure 3.10, these two mappings have smoother reactivity curves.

For the disturbance in coolant flow, seen in figure 3.14 through 3.17, the effect of the bypass mappings is not so prominent, probably due to non-radial dependence of void levels. Another contributing factor might be that the multiple bypasses are connected through the upper plenum, which in turn is connected to the uppermost cells of all bypass channels; the very position where the void is spreading from in case the bypass voids (as have been mentioned earlier, the bypass channels are not connected to each other anywhere but at the upper and lower plenums). The different mappings follow the single bypass solution approximately, with 5 bypass channels slightly below during one time window, and then above in a different one.

The deviations are relatively small, about 5 % around the values for the single by-

pass.

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4.4. REALISTIC ATWS SIMULATION 41

Comparing average bypass densities with the result from the single bypass chan- nel, 3.16 & 3.17 indicates that the 2 bypass channel mapping with equal flow area and 5 bypass channel mapping, seems to be averaging the densities better, giving smoother curves. Interesting is that in figure 3.17, the average density for the 5 channel mapping passes right through the small nick in the curve of the single by- pass transient. This would seem to show that the 5 channel mapping follows the dynamics of the single bypass well, but is not as sensitive to changes in density. A possible answer to this behavior could be that several channels are more adaptable so that each individual channel can more easily follow changes in the local environ- ment, allowing the individual channels to more easily absorb local fluctuations and disturbances, keeping them local and thereby not spreading to, and affecting the whole system. This would in turn give a smoother behavior of the system. Seeing how the bypass represents a large amount of water, rapid fluctuations needs a lot of force to quickly change the density state of the bypass, thus a smoother behavior would seem more physically sound. With such complex models as those present in POLCA-T, it is however difficult to draw precise conclusions of how interactions take part.

4.4 Realistic ATWS Simulation

Looking at figure 3.19 one can observe a difference between the standard model and the heterogeneous model model; k

_eff

decreases quicker for heterogeneous model than the standard model. Figure 3.21 shows a result of importance: the heat released to the containment by heterogeneous model model is slightly lower than with the standard model. The reason for this is that the power generated with the heterogeneous model model is slightly lower than the standard in the region of about 220-300 s. This difference can be explained by a heterogeneity in the boron concentration, seen in figure 3.22, starting at 220 s which makes the heterogeneous model model give negative feedback to reactivity.

Apart from differences mentioned above, the results are similar to the standard cross section model. Other state parameters differ somewhat, but this could very well be from feedback from the previously mentioned effects.

4.5 Uncertainties in Results

All results presented are output from a simulator, which is susceptible to numerical

errors, discretization errors in both time and space, e.t.c. Each new addition to

the code has to pass through rigorous verification and validation, and if possible,

comparison to real world measurements. For the simulations run in this thesis,

focused on properties in the context of injection of borated water, there are no real

world data. Boron pollutes the nuclear core significantly and any such experiment

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heterogeneity model accuracy). Since the synthetic transients in case 1-3 has a pos- itive heterogeneous contribution to k

_eff

, these results are conservative. In the full ABWR ATWS simulations however, the results are not conservative, but will have non-conservative errors in the total k

_eff

of about 100 pcm on average. The standard model is not conservative in under these conditions neither. In fact, according to the internal Westinghouse report, the instances when the heterogeneous model has non-conservative errors, the standard model is usually also non-conservative, but with larger errors.

As was mentioned in 4.3, the results of multiple bypasses are not verified and thus

the results should not be trusted too much for analytical purposes.

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4.5. UNCERTAINTIES IN RESULTS 43

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Chapter 5

Conclusions

The goal of this thesis was to test the application of a newly developed hetero- geneous cross section model in POLCA7 (steady state) for simulation of BWR transients with POLCA-T. The aim of the thesis was to verify the implementation carried over from steady state to the transient regime as intended, and that the model performed adequately and to expectations. The performance of the model in POLCA7, for steady state, has been validated against lattice physics calculations in an internal report at Westinghouse.

To evaluate the new model, various transient problems were solved which demon- strate both the correctness of the implementation, as well as the capability of the model in POLCA-T. Comparison against two reduced models for treatment of cross section heterogeneities proved helpful to confirm the consistency of the results ob- tained with the detailed model. In combination with previous validation work using POLCA7, the reported results provide a good confirmation of the capability of the new model to simulate transients with heterogeneous states present.

The impact of modeling multiple parallel bypass channels have been investigated with non-conclusive results. The general state results, such as total average densi- ties, k

_eff

and boron concentrations, are similar for all bypass mappings, with local variations present in the multiple bypass mappings. Therefore a multiple bypass mapping could be used to gain increased local detail if wanted, but as a general recommendation a single bypass should be used, since it represents the global, av- eraged state and also reduces the complexity of the model, which in turn reduces the risk for numerical anomalies.

The study was completed with the simulation of a realistic anticipated transient without scram (ATWS) event in an ABWR plant, an ATWS event in which the

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(ATWS). The boron solution may distribute differently in the active region and bypass water regions and the assumption of a uniform boron distribution, which is inherent in the original cross section model, may lead to non-conservative results due to under-prediction of core reactivity. For transients involving voiding of by- pass, but without boron injection, the standard model is the most conservative, whereas the new model still emerges as more realistic. It is worth using the reduced models if the cell data file at hand does not include the necessary heterogeneity coefficients. Both these reduced models represent an improvement to the standard method and is a good first approximation in the treatment of heterogeneous density and boron concentration states.

5.1 Suggestions For Further Work

Due to convergence issues and time constraints, a deeper analysis of the relations between multiple internal bypasses and/or the external bypass could not be per- formed. The results found in this thesis points towards a significant impact on reactivity from different mappings, especially in the context of heterogeneously dis- tributed boron. Analysis of the relations would be interesting and most importantly:

which mapping lies closest to the real world physical system? And, how does it affect

the stability and robustness of the model.

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Appendix A

Appendix

A.1 Fuel cost

Approximate fuel costs for 1 kg of enriched Uranium with conversion, enrichment, fuel fabrication and anticipated waste management costs, from WISE uranium project (http://www.wise-uranium.org/) with spot-price (2011-07-27) from UxC nuclear fuel prices.

Table A.1: Calculation of fuel costs for 1 kgU enriched fuel in units of thousand US$.

Nominal Cost: Future Waste Mgmt Cost:

¹

Total Cost:

Natural uranium: 1,091342 0,211911 1,303253

Conversion: 0,087624 - 0,087642

Enrichment: 0,671932 0,071002 0,742934

Fuel Fabrication: 0,460000 - 0,460000

Spent Fuel: - 0,840000 0,840000

Total: 2,310899 1,122913 3,433812

As the table shows, the approximate price for 1 kg ready-to-use, enriched uranium fuel is approximately $2300 ≈ 1800 e (excluding waste management costs).

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Modelling of Heterogeneously Distributed Soluble boron and Coolant Density within a Fuel Assembly